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On Dynamical Behaviour of Fitzhugh-Nagumo Systems Matthias Ringqvist

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On Dynamical Behaviour of Fitzhugh-Nagumo Systems Matthias Ringqvist ISSN: 1401-5617 On Dynamical Behaviour of FitzHugh-Nagumo Systems Matthias Ringqvist Research Reports in Mathematics Number 5, 2006 Department of Mathematics Stockholm University Electronic versions of this document are available at http://www.math.su.se/reports/2006/5 Date of publication: August 23, 2006 Keywords: Dynamical Systems, FitzHugh-Nagumo, Limit Cycles, Bifurcation. Postal address: Department of Mathematics Stockholm University S-106 91 Stockholm Sweden Electronic addresses: http://www.math.su.se/ [email protected] On Dynamical Behaviour of FitzHugh-Nagumo Systems Filosofie licentiatavhandling by Matthias Ringqvist To be presented on the 8th of September 2006 Abstract In this thesis a class of FitzHugh-Nagumo system is studied. By using the theory of Lyapunov coefficient to analyze Hopf and Bautin bifurcation it is shown that at most two limit cycles can bifurcate from the origin in this case. Further it is shown that there exists choices of parameters such that the maximum number of bifurcating limit cycles is obtained and in this case the inner cycle is unstable while the outer is stable. In the certain case when one of the parameters are assumed to be very small, sufficient conditions are presented ensuring the existence of a unique stable limit cycle. By using a theorem by Lefschetz conditions are given ensuring the existence of at least one stable limit cycle. Also some conditions, apart from the well known Bendixson's criteria, on the parameters giving non-existence of limit cycles are presented. The thesis also contains a complete saddle-node bifurcation analysis based on the Center Manifold Theorem as well as a Bogdanov-Takens bifurcation analysis though not equally complete. Finally the thesis contains sufficient conditions for bounded solutions and a discussion about coupled system of the same class of FitzHugh-Nagumo system. 1 A summery with some further discussions 2 Paper I 3 Paper II 4 A case study on weakly coupled FitzHugh-Nagumo oscillators On dynamical Behavior of FitzHugh-Nagumo Systems {A summery with some further discussions In this thesis we deal with a class of polynomial systems dx = Cy Ax(x B)(x λ) + I (1) dt − − − − 8 dy > = "(x δy) < dt − with non-zero parameters:> A; B; C; δ; "; λ and I being an external force which can be a function of t. The main results The main theorem of this thesis is Theorem 5:2 in Paper II. It states that • at most two limit cycles can bifurcate form the origin via Hopf bifurcation. Further, it is shown that there exists parameters such that this upper bound is obtained and that in this case the inner cycle is unstable while the outer cycle together with the origin are stable. The proof uses the theory of Lyapunov coefficients and relies heavily on a theorem by Andronov from the 1960's, see [2]. In the certain case when one of the parameters are assumed to be very • small, singular perturbation theory is applied to give sufficient conditions ensuring existence of a unique stable limit cycle. Conditions on existence of at least one stable limit cycle are given based on • a theorem by Lefschetz. Also some conditions, apart from the well known Bendixson's criteria, on the parameters giving non-existence of limit cycles are presented. A complete saddle-node bifurcation analysis based on the Center Manifold • Theorem as well as a Bogdanov-Takens bifurcation analysis is presented. In the saddle-node bifurcation, a bifurcation curve is provided. It also contains sufficient conditions for bounded solutions and a discussion • about coupled system of the same class of FitzHugh-Nagumo system. Motivation The FitzHugh-Nagumo system described in (1) is commonly referred to as FitzHugh- Nagumo model because it was based on the system independently used by FitzHugh and Nagumo in the beginning of 1960s \to expose to view part of the inner working mechanism of the Hodgkin-Huxley equations", a popular model in study of neuro- physiology since 1952. Later this system has been used in various modelling of bi- ological behavior, ranging from e.g. neurophysiology to modeling of active fibre for cardiac muscle and to implementation of neuron model, see e.g. [3, 7, 5, 8, 15, 20, 24] and references therein. In order to make our motivation clear, we briefly review the development from Hodgkin-Huxley model to FitzHugh-Nagumo simplified model and current research of our interests. Hodgkin-Huxley model for the action potential of a space clamped 1 squid axon is defined by the four dimensional vector field v_ = I 120m3h(v + 115) + 36n4(v 12) + 0:3(v + 10:599) − − v + 25 v 8 m_ = (1 m)Ψ m 4 exp > > − 10 − 18 > > v + 10 v > n_ = (1 n)0:1Ψ n 0:125 exp (2) > − 10 − 80 > <> v h _ h = (1 h)0:07 exp v+30 > − 20 − 1 + exp 10 > > s > Ψ(s) = > exp(s) 1 > − > with variables:> (v; m; n; h) that represent membrane potential, activation of a sodium current, activation of a potassium current, and inactivation of the sodium current and a parameter I that represents injected current into the space-clamped axon. Al- though there are improved models the Hodgkin-Huxley model remains the paradigm for conductance-based models of neural system. FitzHugh was the first investiga- tor to apply qualitative phase-plane methods to understanding the Hodgkin-Huxley model. To make headway in gaining analytic insight, FitzHugh first considered the variables that change most rapidly, viewing all others as slowly varying parameters of the system. In this way he derived a reduced two-dimensional system that could be viewed as a phase plane. Note that the voltage convention adopted by FitzHugh V = vout vin is opposite to what subsequently became entrenched in scientific literature.− From the Hodgkin-Huxley equations FitzHugh noticed that the variables V and m change more rapidly than h and n, at least during certain time intervals. By arbitrarily setting h and n to be constant we can isolate a set of two equations which describe a two-dimensional (V; m) phase plane. The elegance of applying phase- plane methods and reduced systems of equations to this rather complicated problem should not be underestimated. Similar ideas are used nowadays in many biological problems, one of the examples is dynamical analysis of cell cycle regulation, e.g. [25]. In 1961 FitzHugh proposed to demonstrate that the Hodgkin-Huxley model be- longs to a more general class of systems that exhibit the properties of excitability and oscillations. As a fundamental prototype, the van der Pol oscillator was an example of this class, and FitzHugh therefore used it (after suitable modification). A similar approach was developed independently by Nagumo in 1962. FitzHugh proposed the following equations: u_ = c[w + u u3=3 + I]; − w_ = (u a + bw)=c: − − In these equations the variable u represents the excitability of the system and could be identified with voltage (membrane potential in the axon); w is a recovery variable, representing combined forces that tend to return the state of the axonal membrane to rest. Finally I is the applied stimulus that leads to excitation (such as input current), or rectangular pulses. In order to obtain suitable behavior, FitzHugh made the following assumptions about the constants a; b and c: 1 2b=3 < a < 1; 0 < b < 1; b < c2: − 2 Today there are variant formulations of this system, used in a variety of biological oscillations. The meaning of the variables and parameters are different from the original FitzHugh-Nagumo's. The formulation we work on is taken from [8] which is widely used in literature. A short but rather complete description of physiology behind the biological neu- ron and the corresponding derivation of the Hodgkin-Huxley model and its simpli- fied version, the FitzHugh-Nagumo model, can be found in [20]. The thesis attempts to describing different dynamical behavior exhibited by (1), in terms of the six parameters, as complete as possible, and therefore to gaining analytic insight of this widely used system as much as possible. Although the FitzHugh-Nagumo system is a well-studied object (see e.g. [14, 16, 26, 29, 29, 32]), there are several reasons that motivate the current study. It is of mathematical interest to know the number of limit cycles for a polynomial system although we do not have ambition to solve this problem completely. A classical solved example is van der Pol equation without external force to which we know that there is only one stable limit cycle. It is also of interest to give a satisfactory picture of how bifurcation takes place and which parameters (of the six) play role in the bifurcation. According to our knowledge, the parameters are essential in biology and different application areas need different parameters. This is one of the reasons we do not a priori assume the values of some parameters. If the above-mentioned problems are considered to be more difficult, we want to mention that there may not exist a definite answer to some seemingly innocent questions, for example, there is still no definite answer, in some parameter settings, to the question that the solution of FitzHugh-Nagumo equations converges to a fixed point or to a limit cycle [16]. Since the FitzHugh-Nagumo model, defined by (1), is used for investigation of a single neuron, in reality we have to study the interconnection and coupling of neurons. In other words, a chain of such systems will be used in more realistic models for instance, [21]. Therefore we believe that a full description of dynam- ical behavior of the FitzHugh-nagumo system will benefit to understanding more complicated systems based on the FitzHugh-Nagumo system, see \A Case Study on Weakly Coupled FitzHugh-Nagumo Oscillators".
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